"In a finite space there must be a finite number of events" seems to have counterexamples: take events which occur on vanishing small space, then you can fit an infinite number of them in finite space. Think about converging infinite series.
Even simpler: there are an infinite number of real numbers in the interval between 0 and 1. Indeed, between any two real numbers. If spacetime itself isn't quantized then you get an infinite number of possible events, even within a finite hypervolume of spacetime.
